Yes. THEOplayer supports vertical video playback, which is also known as portrait mode.
More specifically, you can configure the aspect ratio of the container which holds the THEOplayer video player instance.
Of course, you can also configure your container to respect any other aspect ratio, like 5:4, which is employed by Instagram.
A demo is available at https://cdn.theoplayer.com/demos/vertical/index.html. This demo provides a sample implementation which allow you to switch between popular viewing modes. This demo is best viewed on a mobile browser.